Persistence and global stability in discrete models of pure-delay nonautonomous Lotka–Volterra type

Abstract

Consider the persistence and the global asymptotic stability of the following discrete model of pure-delay nonautonomous Lotka–Volterra type: N i(p+1)=N i(p) exp c i(p)− ∑ j=1 n ∑ l=0 m a ij l(p)N j(p−k l) , p=0,1,2,…, 1⩽i⩽n, N i(p)=N i,p⩾0,p⩽0, andN i,0>0,1⩽i⩽n, where each c i ( p) and a ij l ( p) are bounded for p⩾0 and inf p⩾0 ∑ l=0 m a ii l(p) >0, a ij l(p)⩾0,i⩽j⩽n, 1⩽i⩽n, andk l⩾0,1⩽l⩽m. In this paper, for the above discrete system of pure-delay type, by improving the former work [J. Math. Anal. Appl. 273 (2002) 492–511] which extended the averaged condition offered by S. Ahmad and A.C. Lazer [Nonlinear Anal. 40 (2000) 37–49], we offer conditions of persistence, and considering a Lyapunov-like discrete function to the above discrete system, we establish sufficient conditions of global asymptotic stability.